Q3 will work upside down; the hFE will be low but it will still work as a transistor (it will actually saturate at a lower voltage but that is besides the point).
Why doesn't this work? Or better still why do relaxation oscillators work - the simple answer is hysteresis. Your circuit has no hysteresis therefore it operates in a linear fashion and finds a perfectly likable equilibrium with about 0.7 volts across R2.
What would have been preferred is for Q3 to act like a trigger - once activated it stays activated until the voltage across C1/R2 has dropped (maybe) to 50%. Then the whole process will start again with C1 rapidly charging then triggering Q2 which then discharges C1/R2 thus an oscillator is created. Remember that Q3 can work upside down and the base-collector region is the new base-emitter region.
Sorry, it won't happen like that in a linear circuit - it just finds a nice equilibrium and sits there. Relation oscillators need hysteresis! Here's the archetypal op-amp relation oscillator: -
R3 and R2 provide positive feedback and this creates hysteresis. Here's a useful oscillator to try: -
It's called an astable multivibrator but if you want some other options hit this link.
Imposing a timestep does not make it faster, and if you need speed and accuracy, I'm afraid that's not very possible.
There are some TVSs in series, quite a lot of them, which can be replaced by one TVS with n=X (= the number of series elements). If we're at it, m=Y will set the number of parallel devices. Note that only m is valid for RLCs, n only for diodes. This can simply be added after the instance name. For example, two series and three parallel 4148s will look like 1N4148 n=2 m=3. They will not count towards the final node count because they're expanded internally, but they will count towards the computation, since LTspice still has to compute the presence of 6 diodes.
For the floating V5, if that is one offending element (which could be, since LTspice even specifies in its manual that current sources are recommended over their voltage counterparts and voltage sources should be tied to ground for best performance), the cure is simple: add Rser=1m. This will transform, also internally, the voltage surce into its Norton equivalent, thus improving convergence.
You can also combine series RL with L Rser=x, same for caps, same for parallel and/or series combinations. Same explanation as for the TVS.
As for settings, you're better off making trtol=3..7 instead of the others. There will be a (minor, -ish) speedup, depending on your hardware and schematic, while the precision doesn't have tham much of an impact as gmin, reltol and abstol have.
There is one more thing that puzzles me: in one of the comments, someone suggests using current sources instead of optocouplers, and you say you tried. This makes me think accuracy, or keeping to a quasi-real setup, is not that important to you, which means you could simplify theLC filter after V5 into it's simple LC lowpass (i.e. don't make it a symmetric filter), but the biggest simplification can be done to the whole bridge and its control circuitry: you can simply use some G (or E) sources driving the native switch SW. The SW may need some anti-parallel diodes. Speaking of which, you can also replace the diodes with the idealized version, having .model D D Vfwd=0.7 Vrev=1k Ron=0.1 Roff=10Meg epsilon=100m revepsilon=50m, or Vfwd=0.5 for Shottky. I see two anti-parallel diodes, those could be replaced by only one diode with Vfwd=Vrev. Zeners also with Vrev=X. Of course, all these imply using an idealized, or a behavioural approach to all your schematic and, while it's very plausible and used for quick tests, you should not forget that the downside is the unrealistic results, even when modeled with great care. You could get good results, but they shouldn't be relied on, as even a schematic made with "real" elements is only a SPICE simulation using models that, themselves, are approximation of real-life cases. Of course, ultimately, it falls on you to choose your way.
Best Answer
The circuit is too symmetrical and does not "run" on a simulator. You have introduced some asymmetry with the base resistors, but this is not enough, as the transistors are well saturated with small tolerances. I changed C1 to 20uF and the circuit started.